Basic algebra. Volume II

681 Pages·2009·18.106 MB·English

by  Jacobson, Nathan

BASIC
ALGEBRA
II
Second Edition
Nathan Jacobson
Yale University
Dover Publications, Inc.
Mineola, New York
To Mike and Polly
Copyright
Copyright © 1980, 1989 by Nathan Jacobson
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2009, is an unabridged republication of the 1989 second edition of the work originally
published by W. H. Freeman and Company, San Francisco, in 1980.
Library of Congress Cataloging-in-Publication Data
Jacobson, Nathan, 1910–1999.
Basic algebra / Nathan Jacobson. — Dover ed.
p. cm.
Originally published: 2nd ed. San Francisco : W.H. Freeman, 1985–1989.
ISBN-13: 978-0486-47189-1 (v. 1)
ISBN-10: 0-486-47189-6 (v. 1)
ISBN 13: 978-0-486-47187-7 (v. 2)
ISBN-10 0-486-47187-X (v. 2)
1. Algebra I. Title.
QA154.2.J32 2009
512.9—dc22
2009006506
Manufactured in the United States by Courier Corporation
47187X01
www.doverpublications.com
Contents
Contents of Basic Algebra I
Preface
Preface to the First Edition
INTRODUCTION
0.1 Zorn’s lemma
0.2 Arithmetic of cardinal numbers
0.3 Ordinal and cardinal numbers
0.4 Sets and classes
References
1 CATEGORIES
1.1 Definition and examples of categories
1.2 Some basic categorical concepts
1.3 Functors and natural transformations
1.4 Equivalence of categories
1.5 Products and coproducts
1.6 The horn functors. Representable functors
1.7 Universals
1.8 Adjoints
References
2 UNIVERSAL ALGEBRA
2.1 Ω-algebras
2.2 Subalgebras and products
2.3 Homomorphisms and congruences
2.4 The lattice of congruences. Subdirect products
2.5 Direct and inverse limits
2.6 Ultraproducts
2.7 Free Ω-algebras
2.8 Varieties
2.9 Free products of groups
2.10 Internal characterization of varieties
References
3 MODULES
3.1 The categories R-mod and mod-R
3.2 Artinian and Noetherian modules
3.3 Schreier refinement theorem. Jordan-Hölder theorem
3.4 The Krull-Schmidt theorem
3.5 Completely reducible modules
3.6 Abstract dependence relations. Invariance of dimensionality
3.7 Tensor products of modules
3.8 Bimodules
3.9 Algebras and coalgebras
3.10 Projective modules
3.11 Injective modules. Injective hull
3.12 Morita contexts
3.13 The Wedderburn-Artin theorem for simple rings
3.14 Generators and progenerators
3.15 Equivalence of categories of modules
References
4 BASIC STRUCTURE THEORY OF RINGS
4.1 Primitivity and semi-primitivity
4.2 The radical of a ring
4.3 Density theorems
4.4 Artinian rings
4.5 Structure theory of algebras
4.6 Finite dimensional central simple algebras
4.7 The Brauer group
4.8 Clifford algebras
References
5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS
5.1 Representations and matrix representations of groups
5.2 Complete reducibility
5.3 Application of the representation theory of algebras
5.4 Irreducible representations of S
n
5.5 Characters. Orthogonality relations
5.6 Direct products of groups. Characters of abelian groups
5.7 Some arithmetical considerations
5.8 Burnside’s paqb theorem
5.9 Induced modules
5.10 Properties of induction. Frobenius reciprocity theorem
5.11 Further results on induced modules
5.12 Brauer’s theorem on induced characters
5.13 Brauer’s theorem on splitting fields
5.14 The Schur index
5.15 Frobenius groups
References
6 ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS
6.1 Additive and abelian categories
6.2 Complexes and homology
6.3 Long exact homology sequence
6.4 Homotopy
6.5 Resolutions
6.6 Derived functors
6.7 Ext
6.8 Tor
6.9 Cohomology of groups
6.10 Extensions of groups
6.11 Cohomology of algebras
6.12 Homological dimension
6.13 Koszul’s complex and Hilbert’s syzygy theorem
References
7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND
NOETHERIAN RINGS
7.1 Prime ideals. Nil radical
7.2 Localization of rings
7.3 Localization of modules
7.4 Localization at the complement of a prime ideal. Local-global relations
7.5 Prime spectrum of a commutative ring
7.6 Integral dependence
7.7 Integrally closed domains
7.8 Rank of projective modules
7.9 Projective class group
7.10 Noetherian rings
7.11 Commutative artinian rings
7.12 Affine algebraic varieties. The Hilbert Nullstellensatz
7.13 Primary decompositions
7.14 Artin-Rees lemma. Krull intersection theorem
7.15 Hilbert’s polynomial for a graded module
7.16 The characteristic polynomial of a noetherian local ring
7.17 Krull dimension
7.18 I-adic topologies and completions
References
8 FIELD THEORY
8.1 Algebraic closure of a field
8.2 The Jacobson-Bourbaki correspondence
8.3 Finite Galois theory
8.4 Crossed products and the Brauer group
8.5 Cyclic algebras
8.6 Infinite Galois theory
8.7 Separability and normality
8.8 Separable splitting fields
8.9 Kummer extensions
8.10 Rings of Witt vectors
8.11 Abelian p-extension
8.12 Transcendency bases
8.13 Transcendency bases for domains. Affine algebras
8.14 Luroth’s theorem
8.15 Separability for arbitrary extension fields
8.16 Derivations
8.17 Galois theory for purely inseparable extensions of exponent one
8.18 Tensor products of fields
8.19 Free composites of fields
References
9 VALUATION THEORY
9.1 Absolute values
9.2 The approximation theorem
9.3 Absolute values on and F(x)
9.4 Completion of a field
9.5 Finite dimensional extensions of complete fields. The archimedean case
9.6 Valuations
9.7 Valuation rings and places
9.8 Extension of homomorphisms and valuations
9.9 Determination of the absolute values of a finite dimensional extension field
9.10 Ramification index and residue degree. Discrete valuations
9.11 Hensel’s lemma
9.12 Local fields
9.13 Totally disconnected locally compact division rings
9.14 The Brauer group of a local field
9.15 Quadratic forms over local fields
References
10 DEDEKIND DOMAINS
10.1 Fractional ideals. Dedekind domains
10.2 Characterizations of Dedekind domains
10.3 Integral extensions of Dedekind domains
10.4 Connections with valuation theory
10.5 Ramified primes and the discriminant
10.6 Finitely generated modules over a Dedekind domain
References
11 FORMALLY REAL FIELDS
11.1 Formally real fields
11.2 Real closures
11.3 Totally positive elements
11.4 Hilbert’s seventeenth problem
11.5 Pfister theory of quadratic forms
11.6 Sums of squares in R(x ,…,x ), R a real closed field
1 n
11.7 Artin-Schreier characterization of real closed fields
References
INDEX